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**TESTS OF STATISTICAL SIGNIFICANCE**

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**STATISTICAL INFERENCE**

TRUE STATE OF AFFAIRS EFFECTIVE NOT EFFECTIVE True positive(correct) False positive(type I / alpha error) Conclusion of study False negative(type II /beta error) True negative (correct)

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**AVAILABLE TESTS STANDARD ERROR TESTS**

making use of limit/choice of 95% level of confidence to declare any value falling outside the 95% confidence interval or two standard errors to be statistically significant These include MEANS /,PROPORTIONS OF ONE SAMPLE DIFEERENCE BETWEEN MEANS / PROPORTIONS OF TWO SAMPLES

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**FORMULAE FOR ESTIMATION OF STANDARD ERROR(SE) OF SAMPLE**

1. SE of sample mean= SD/ √n 2. SE of sample proportion(p) = √pq/n 3. SE of difference between two means[SE(d)]=√SD1/ n1+ SD2/n2 4. SE of difference between two proportions= √p1q1/n1+ p2q2/n2

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REFERENCE RANGES The 95% limits( REFER TO 2 Std deviations on either side of mean) and are referred to as REFERENCE RANGE For many biological variables they define what is regarded as the NORMAL RANGE OF THE NORMAL DISTRIBUTION

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**THE AIM OF A STATISTICAL TEST**

To reach a scientific decision (“yes” or “no”) on a difference (or effect), on a probabilistic basis, on observed data.

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SIGNIFICANCE TESTING GOAL: see if observed test result/ hypothesized difference is likely to be due to chance PRINCIPLE: Process of significance testing or statistical inference(HA) based on principle of relating the observed findings to the hypothetical true state of affairs(H0)

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**HOW TO DECIDE WHEN TO REJECT THE NULL HYPOTHESIS?**

H0 rejected using reported p value arbitrarily chosen to be REMEMBER THE REMAINING 5% OF DATA LEFT OUT WHEN USING 95 % CONFIDENCE INTERVAL IN STANDARD ERROR APPROACH p-value = probability that our result (e.g. a difference between proportions or a RR) or more extreme values could be observed under the null hypothesis

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**STEPS IN SIGNIFICANCE TESTING**

1. generating hypothesis NULL HYPOTHESIS VS ALTERNATE HYPOTHESIS 2. determine probability of error P VALUE OF SAY 0.05 OR 0.01 (if more confident) 3. choose appropriate statistical test; calculate test statistic 4. obtain the p value for data of the particular sample statistic 5. describe inference and conclusion

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**Types of errors Truth Decision based on the p value**

No diff Diff Decision based on the p value No diff Diff H0 is “true” but rejected: Type I or error H0 is “false” but not rejected: Type II or error

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**PROBABILITY OF TYPE I ERROR AND LEVEL OF α**

10% 0.1 5% 0.05 1% 0.01 0.1% 0.001 0% .000

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**CHOOSING APPROPRIATE STATISTICAL TESTS**

Whether comparison is between independent or related groups Whether proportion or means are being compared Whether more than two groups are being compared

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**Examples of Sample Statistics:**

Single population mean Single population proportion Difference in means (ttest) Difference in proportions (Z-test),x2 test Odds ratio/risk ratio Correlation coefficient Regression coefficient … It turns out that if you were to go out and sample many, many times, most sample statistics that you could calculate would follow a normal distribution. What are the 2 parameters (from last time) that define any normal distribution? Remember that a normal curve is characterized by two parameters, a mean and a variability (SD) What do you think the mean value of a sample statistic would be? The standard deviation? Remember standard deviation is natural variability of the population Standard error can be standard error of the mean or standard error of the odds ratio or standard error of the difference of 2 means, etc. The standard error of any sample statistic.

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**COMMON STATISTICAL TESTS**

DESIGN NATURE OF VARIABLES STAISTICAL TEST STATISTIC DERIVED 2 independent groups Qualitative (nominal) Qunatitative (continuous) Chi –square test Student t -test ×2 t 2 related groups Paired t -test More than 2 independent groups ANOVA F

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**MANY STATISTICS FOLLOW NORMAL (OR T-DISTRIBUTIONS)…**

Means/difference in means T-distribution for small samples Proportions/difference in proportions Regression coefficients Natural log of the odds ratio

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**PARAMETRIC TESTS(t,z and f tests)**

Test hypothesis about means or variances Their hypothesis refer to population ( population mean z or t tests) Population variance (f tests) Their hypothesis concern interval / ratio scale data (wt , BP, IQ) Assume population data to be normally distributed

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**NON PARAMETRIC TESTS –(chi sq test)**

Do not test hypothesis concerning parameters Do not assume normal distribution of data Used to test nominal/ordinal data Less powerful than parametric tests

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**CHI –SQUARE TEST- essentials**

Used to find out whether observed difference between proportions of events in 2 or more groups can be considered statistically significant Non parametric test Use qualitative ,discrete data in proportions or frequencies (not in %) Statistic calculated is chi square(x2)

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**STEPS OF APPLYING CHI- SQUARE TEST(x2)**

An assumption of no difference is made which is then proved or disproved with the help of x2 test . (THE NULL HYPOTHESIS) Fix a level of significance (0.05)-the p –value Enter study data in the table ,(observed frequency (O) Calculate expected frequency for each cell(E) Formula of x2 value for each cell =(O-E)2/E Expected frequency = (RTxCT/GT) Add up results of all cells x2calcul=∑(O-E)2/ E Df = (C-1)x(R-1) its 1 in a 2x2 table Read x2 table for corresponding p- value Compare this p- value with level of significance (0.05) and decide fate of the alternate or research hypothesis

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**Is the daily use of library facility associated with shorter distance from hostels?**

Distance from lib Used lib in evening Not used lib in evening total Less than 1 km (O)51 (E=44.4) (O)29 (E=35.6) 80 1 km or more (0)35 (E=41.6) 86 (0)40 (E=33.4) 69 75 155 Expected values are calculated on the Basis of Ho i.e there is no difference in lib utilization time between 2 groups of students ///////////////// =

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**P – value in sample statistic is < 0.05 so **

X 2= ( )2 + ( ) 2 + ( )2 + ( )2 ====== 4.55 X 2 at 2 df = 4.55 ( p value is < 0.05 in table) for P- value at 0.05, X 2 in table is 3.84 at 2 df P – value in sample statistic is < 0.05 so The results are significant and H0 is REJECTED

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**STUDENT’S T-TEST Applied for numerical data (mean values)**

Normal distribution of variable assumed Random sampling Steps Calculate t- value (from data) Choose level of significance, p- value 0.05 Determine df (sum of 2 samples sizes minus 2 ) Find t value for LOS decided from table, if calculated t value is > than table value , Ho is rejected .

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**HOW TO CALCULATE t- VALUE**

Calculate means of any 2 groups/samples(X1,X2) Calculate their difference(X1- X2) Calculate standard deviation (SD)of each group Calculate standard error(SE) for each SD/√n) The t- value= X1-X2/ √ SD1/n1+ √ SD2/n2

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**example Delivery outcome N Mean height SD Normal B Wt 60 156cm 3.1**

Low B Wt 52 152cm 2.8

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Ho= there is no difference in mean heights of the two groups--- difference may be by chance but acceptable level of significance is 0.05 t= √ x1-x = Sd12/n1 +sd22/n √3.12/ /52 = = 0.56 Value of t at df=110 at 0.05 LOS is 1.98(< 3.6) So difference is significant and Ho is rejected

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CORRELATION Establish and quantify strength and direction of relationship between 2 variables by “Correlation coefficient” symbol : r Range -1 or + 1 Size of coefficient: signifies strength Sign of coefficient : signifies direction 2 types of correlation coefficients used Pearson correlation coefficient: interval/ratio scale Spearman correlation coefficient: ordinal scale data

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**Correlation Coefficient**

The population correlation coefficient ρ (rho) measures the strength of the association between the variables The sample correlation coefficient r is an estimate of ρ and is used to measure the strength of the linear relationship in the sample observations

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**Scatter Plots and Correlation**

A scatter plot (or scatter diagram) is used to show the relationship between two variables Correlation analysis is used to measure strength of the association (linear relationship) between two variables Only concerned with strength of the relationship No causal effect is implied

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**Scatter Plot Examples Linear relationships Curvilinear relationships y**

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**Scatter Plot Examples Strong relationships Weak relationships y y x x**

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Scatter Plot Examples No relationship y x y x

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**Examples of Approximate r Values**

y y y x x x r = -1 r = -.6 r = 0 y y x x r = +.3 r = +1

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**Significance Test for Correlation**

Hypotheses H0: ρ = 0 (no correlation) HA: ρ ≠ 0 (correlation exists) Test statistic (with n – 2 degrees of freedom)

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REGRESSION Means change in measurements of a variable character on the positive or negative side beyond the mean For strongly correlated variables , value of dependent variable can be predicted from the values of the independent variable Simple linear regression Multiple linear regression Logistic regression

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**HOPE FEELING READY TO LAUNCH THE NEXT RESEARCH HYPOTHESIS WITH**

95% CONFIDENCE LEVEL AND < 5% P-VALUE FOR THE TEST STATISTIC or < 5% LEVEL OF TYPE 1 ERROR or <20% LEVEL OF TYPE II ERROR AND 80% POWER OF STUDY TO PICK UP THE REAL DIFFERENCE

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AM 680.01 Recitation 2/10/11.

AM 680.01 Recitation 2/10/11.

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